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NormalToricVarieties :: sheaf(NormalToricVariety,Module)

sheaf(NormalToricVariety,Module) -- make a coherent sheaf

Synopsis

Description

The category of coherent sheaves on a normal toric variety is equivalent to the quotient category of finitely generated modules over the total coordinate ring by the full subcategory of torsion modules with respect to the irrelevant ideal. In particular, each finitely generated module over the total coordinate ring corresponds to coherent sheaf on the normal toric variety and every coherent sheaf arises in this manner. For more information, see Subsection 5.3 in Cox-Little-Schenck's Toric Varieties.

Free modules correspond to reflexive sheaves.

i1 : PP3 = toricProjectiveSpace 3;
i2 : F = sheaf (PP3, (ring PP3)^{{1},{2},{3}})

          1            1            1
o2 = OO    (1) ++ OO    (2) ++ OO    (3)
       PP3          PP3          PP3

o2 : coherent sheaf on PP3
i3 : FF7 = hirzebruchSurface 7;
i4 : G = sheaf (FF7, (ring FF7)^{{1,0},{0,1}})

          1                1
o4 = OO     (1, 0) ++ OO     (0, 1)
       FF7              FF7

o4 : coherent sheaf on FF7

See also